Covering Symmetric Sets of the Boolean Cube by Affine Hyperplanes

Alon and F\"uredi (European J. Combin., 1993) proved that any family of hyperplanes that covers every point of the Boolean cube $\{0,1\}^n$ except one must contain at least $n$ hyperplanes. We obtain two extensions of this result, in characteristic zero, for hyperplane covers of symmetric sets of the Boolean cube (subsets that are closed under permutations of coordinates), as well as for `polynomial covers' of `weight-determined' sets of `strictly unimodal uniform' (SU$^2$) grids. As a main tool for solving our problems, we give a combinatorial characterization of (finite-degree) Zariski (Z-) closures of symmetric sets of the Boolean cube -- the Z-closure of a symmetric set is symmetric. In fact, we obtain a characterization that concerns, more generally, weight-determined sets of SU$^2$ grids. However, in this generality, our characterization is not of the Z-closures -- unlike over the Boolean cube, the Z-closure of a weight-determined set need not be weight-determined. We introduce a new closure operator exclusively for weight-determined sets -- the `(finite-degree) Z*-closure' -- defined to be the maximal weight-determined set in the Z-closure. (This coincides with the Z-closure over the Boolean cube, for symmetric sets.) We obtain a combinatorial characterization of the finite-degree Z*-closures of weight-determined sets of an SU$^2$ grid. This characterization may also be of independent interest. Indeed, as further applications, we (i) give an alternate proof of a lemma by Alon et al. (IEEE Trans. Inform. Theory, 1988), and (ii) characterize the `certifying degrees' of weight-determined sets. Over the Boolean cube, our above characterization can also be derived using a result of Bernasconi and Egidi (Inf. Comput., 1999). However, our proof is independent of this result, works for all SU$^2$ grids, and could be regarded as being more combinatorial.


Introduction and overview
We will work over the field R. For any two integers a, b ∈ Z, a ≤ b, by abuse of notation, we will denote the interval of all integers between a and b by [a, b]. Further, the interval of integers [1, n] will also be denoted by [n]. By a uniform grid, we mean a finite grid of the form [0, k 1 −1]×· · ·×[0, k n −1], for some k 1 , . . . , k n ∈ Z + . (This means for each i ∈ [n], the values taken by the points of the grid in the i-th coordinate are equispaced.) Consider a uniform grid G = [0, k 1 − 1] × · · · × [0, k n − 1]. For any x = (x 1 , . . . , x n ) ∈ G, define the weight of x as wt(x) = i∈[0,n] x i . We define a subset S ⊆ G to be weight-determined if x ∈ S, y ∈ G, wt(y) = wt(x) =⇒ y ∈ S.
Let N = i∈[0,n] (k i − 1). It follows that there is a one-to-one correspondence between weightdetermined sets of G and subsets of [0, N ] -a subset E ⊆ [0, N ] corresponds to a weight-determined set E ⊆ G, defined as the set of all elements x ∈ G satisfying wt(x) ∈ E. When E = {j}, a singleton set, we will denote E by j. Further, we will freely use this one-to-one correspondence and identify the weight-determined set E with E without mention, whenever convenient. This identification will be clear from the context. In addition, for E ⊆ [0, N ], we will denote |E| by G E . It is immediate that G j = G N −j , for all j ∈ [0, N ]. We say the uniform grid G is strictly unimodal if > · · · > G N .
We will abbreviate the term 'strictly unimodal uniform grid' by 'SU 2 grid'. The SU 2 condition on uniform grids is not very restrictive; there are enough interesting uniform grids which are SU 2 . For instance, the uniform grid [0, k − 1] n is SU 2 . There is a simple characterization of the SU 2 grids in terms of their dimensions, given by Dhand [Dha14] (stated in our work as Theorem 2.3).
We will stick with the above notations whenever we consider uniform grids. Further, we will assume throughout that k i ≥ 2, for all i ∈ [n].
The Boolean cube setting. Consider the Boolean cube {0, 1} n . In this case, for any x ∈ {0, 1} n , the weight wt(x) is equal to |x|, the Hamming weight of x, and {0,1} n j = n j , j ∈ [0, n]. It is easy to check that {0, 1} n is strictly unimodal. We define S ⊆ {0, 1} n to be symmetric if S is closed under permutations of coordinates. It follows quite trivially that a subset is weight-determined if and only if it is symmetric. This is not true though for any other uniform grid. The Boolean cube {0, 1} n ⊆ [0, k 1 − 1] × · · · × [0, k n − 1] is clearly symmetric but not weight-determined, if any of the k i -s is at least 3. Also without loss of generality, if k i < k j for some i, j ∈ [n], i = j, then trivially the grid [0, k 1 − 1] × · · · × [0, k n − 1] is weight-determined but not symmetric. In the case all the k i -s are equal, every weight-determined set is indeed symmetric.

Our hyperplane and polynomial covering problems
The term hyperplane covering problem is commonly used in the literature (see Subsection 1.2) to refer to any problem of finding the minimum number of hyperplanes covering a finite set in a finite-dimensional vector space (over a field) while satisfying some conditions. Borrowing this terminology, we use the term polynomial covering problem to refer to any problem of finding the minimum degree of a polynomial covering 2 a finite set in a finite-dimensional vector space (over a field) while satisfying some conditions. We will denote the polynomial ring R[X 1 , . . . , X n ] by R[X]. Further, for any α ∈ N n , we denote the monomial X α 1 1 · · · X αn n by X α .
In this work, we will consider extensions of this result to weight-determined sets of a uniform grid. Let G be a uniform grid. For a weight-determined set E, where E [0, N ], we say a family of hyperplanes H in R n is Let HC G (E) and PHC G (E) denote the minimum sizes of a nontrivial hyperplane cover and a proper hyperplane cover respectively, for a weight-determined set E, E [0, N ]. In the case G = {0, 1} n , we will instead use the notations HC n (E) and PHC n (E) respectively.
For any P (X) ∈ R[X], we denote by Z G (P ), the set of all a ∈ G such that P (a) = 0. For a weight-determined set E, where E [0, N ], we say a polynomial P (X) ∈ R[X] is • a nontrivial polynomial cover of E if E ⊆ Z G (P ) = G.
• a proper polynomial cover of E if E ⊆ Z G (P ) and j ⊆ Z G (P ), for every j ∈ [0, N ] \ E.
Let PC G (E) and PPC G (E) denote the minimum degree of a nontrivial polynomial cover and a proper polynomial cover respectively, for a weight-determined set E, E [0, N ]. In the case G = {0, 1} n , we will instead use the notations PC n (E) and PPC n (E).
We are interested in the following problems.  3 In [AF93], this result was proven true over any field.
All the above variants of hyperplane and polynomial covers coincide for the symmetric set [N ], and hence in our notation, Alon and Füredi [AF93] proved the following.
In this work, we will give combinatorial characterizations of PC G (E) and PPC G (E), for all E [0, N ], for an SU 2 grid G. The characterization of HC G (E) and PHC G (E), for all E [0, N ], for an SU 2 grid G is left open. However, in the Boolean cube setting, we will obtain HC n (E) = PC n (E) and PHC n (E) = PPC n (E), for all E [0, n]. In short, we will solve Problem 1.2 (b) for SU 2 grids, and further solve Problem 1.2 (a) for the Boolean cube. Problem 1.2 (a) for SU 2 grids is open.
Our combinatorial characterizations that answer Problem 1.2 (b) for SU 2 grids are as follows.
Further, we answer Problem 1.2 (a), in the Boolean cube setting, as follows.

Related work
Alon and Füredi [AF93] mention that their hyperplane covering problem was extracted by Bárány from Komjáth [Kom94]. Some of the extensions and variants studied subsequently (over R) are as follows.
• Linial and Radhakrishnan [LR05] gave an upper bound of ⌈n/2⌉ and a lower bound of Ω( √ n) on the minimum size of essential hyperplane covers of the Boolean cube -a family of hyperplanes H is an essential hyperplane cover if H is a minimal family covering {0, 1} n , and each coordinate is influential for a linear polynomial representing some hyperplane in H. Saxton [Sax13] later gave a tight bound of n + 1 for this problem, in the case where the linear polynomials representing the hyperplanes are restricted to be of the form i∈[0,n] a i X i − b, where a i ≥ 0 for all i ∈ [n], and b ≥ 0. Recently, Yehuda and Yehudayoff [YY21] improved the lower bound in the unrestricted case to Ω(n 0.52 ).
• Kós, Mészáros and Rónyai [KR12] introduced the following multiplicity extension: given a finite grid S = S 1 × · · · × S n with each (S i , m i ) being a multiset such that 0 ∈ S i , m i (0) = 1, find the minimum number of hyperplanes such that each point s ∈ S \ {0} is covered by at least i∈[n] m i (s i ) − n + 1 hyperplanes and the point 0 is not covered by any hyperplane. They gave a tight lower bound of i∈[n] m i (S i ) − n. This bound is in fact true over any field.
• Aaronson, Groenland, Grzesik, Johnston and Kielak [AGG + 20] considered exact hyperplane covers of subsets of the Boolean cube -a family of hyperplanes H is an exact hyperplane cover of a subset S ⊆ {0, 1} n if H∈H H ∩ {0, 1} n = S. They obtained tight bounds on the minimum size of exact hyperplane covers for subsets S with |{0, 1} n \ S| ≤ 4, and asymptotic bounds for general subsets.
• Clifton and Huang [CH20] considered another multiplicity version of the hyperplane covering problem: find the least number of hyperplanes required to cover all points of the Boolean cube except the origin k times and not cover the origin at all. They proved the tight bound of n + 1 and n + 3, for k = 2 and k = 3 respectively, and gave a lower bound of n + k + 1 for k ≥ 4. Sauermann and Wigderson [SW20] considered the polynomial version of this problem: find the least degree of a polynomial that vanishes at all points of the Boolean cube, except the origin, k times and vanishes at the origin j times, for some j < k. They gave the tight bounds n + 2k − 3 for j ≤ k − 2, and n + 2k − 2 for j = k − 1.
Several more variants and extensions, in particular over positive characteristic, have appeared in the literature both before and after Alon and Füredi [AF93] -in Jamison [Jam77], Brouwer [BS78], Ball [Bal00], Zanella [Zan02], Ball and Serra [BS09], Blokhuis [BBS10], and Bishnoi, Boyadzhiyska, Das and Mészáros [BBDM21], to quote a few. For a detailed history of the hyperplane covering problems as well as the polynomial method, see, for instance, the nice introduction in [BBDM21].

Finite-degree Z-closures and Z*-closures, and polynomial covering problems
The finite-degree Zariski closure was defined by Nie and Wang [NW15] towards obtaining a better understanding of the applications of the polynomial method to combinatorial geometry. This is a closure operator 4 and has been studied implicitly even earlier (see for instance, Wei [Wei91], Heijnen and Pellikaan [HP98], Keevash and Sudakov [KS05], and Ben-Eliezer, Hod and Lovett [BHL12]). We will abbreviate the term 'Zariski closure' by 'Z-closure'.

Finite-degree Z-closures and Z*-closures
Let G be a uniform grid. For any d ∈ [0, N ] and any S ⊆ G, the degree-d Z-closure of S, denoted by Z-cl G,d (S), is defined to be the common zero set, in G, of all polynomials that vanish on S, and have degree at most d. 5 In the case G = {0, 1} n , we will instead use the notation Z-cl n,d (S).
The finite-degree Z-closures are relevant to us in the context of our polynomial covering problems. We are interested in polynomial covering problems that impose conditions on weight-determined sets, and thus, we are interested in finite-degree Z-closures of weight-determined sets. Intriguingly, the finite-degree Z-closure of a weight-determined set need not be weight-determined.
We will circumvent this issue by introducing a new closure operator, defined exclusively for weight-determined sets. Let G be a uniform grid. For any d ∈ [0, N ] and E ⊆ [0, N ], we define the degree-d Z*-closure of E, denoted by Z*-cl G,d (E), to be the maximal weight-determined set contained in Z-cl G,d (E). In other words, Z-cl G,d (E) is defined by the implications: It follows easily that Z*-cl G,d is a closure operator 6 ; we will prove this in Section 2.
Notation. By definition, it is clear that the finite-degree Z*-closure is a weight-determined set. So, whenever convenient, we will use our identification of weight-determined sets with subsets of [0, N ] while describing these closures. For E ⊆ [0, N ], the notation Z*-cl G,d (E) will denote the Z*-closure as a subset of G, and the notation Z*-cl G,d (E) will denote the Z*-closure as a subset of [0, N ]. Similar 'double notations' would also apply to Z-closures of symmetric sets of the Boolean cube.
The relevance of the finite-degree Z*-closures to our polynomial covering problems is captured by the following simple lemma, which is quite immediate from the definitions.

Combinatorial characterization of finite-degree Z*-closures
We will proceed to give a combinatorial characterization of finite-degree Z*-closures. We need a couple of set operators to proceed.
We are now ready to state our main theorem: our combinatorial characterization of finite-degree Z*-closures of weight-determined sets of an SU 2 grid.
In addition, we can characterize when the finite-degree Z*-closure of a weight-determined set E is equal to E or [0, N ].
This would complete the proof of Theorem 1.4, thus completing our solution to Problem 1.2 (b) for SU 2 grids.

Finite-degree h-closures and h*-closures, and hyperplane covering problems
To better understand the difference between the hyperplane and polynomial covering problems, we introduce another new closure operator, which we call the finite-degree h-closure, defined using polynomials representing hyperplane covers. Let H n be the set of all polynomials in R[X] := R[X 1 , . . . , X n ] which are products of polynomials of degree at most 1. Let G be a uniform grid. For any d ∈ [0, N ] and any S ⊆ G, we define the degree-d h-closure of S, denoted by h-cl G,d (S), to be the common zero set, in G, of all polynomials in H n that vanish on S and have degree at most d.
In the case G = {0, 1} n , we will instead use the notation h-cl n,d (S).
Note that we do not know if the finite-degree hyperplane closure of every weight-determined set of G is weight-determined. So, akin to the definition of finite-degree Z*-closures, for any d ∈ [0, N ] and E ⊆ [0, N ], we define the degree-d h*-closure of E, denoted by h*-cl G,d (E), to be the maximal weight-determined set contained in h-cl G,d (E). These closures are relevant to us in the context of our hyperplane covering problems due to the following observation, which is immediate from the definitions.

The Boolean cube setting: characterizing h-closures
We will characterize the finite-degree h-closures of all symmetric sets of the Boolean cube for all degrees; in fact, we make the intriguing observation that these coincide with the finite-degree Zclosures.
Notation. It is easy to see that the finite-degree h-closure of a symmetric set of a Boolean cube is So once again, we will use the identification between symmetric sets of {0, 1} n and subsets of [0, n]. For E ⊆ [0, n], the notation h-cl n,d (E) will denote the h-closure as a subset of {0, 1} n , and the notation h-cl n,d (E) will denote the h-closure as a subset of [0, n]. We will also follow the same convention if we use the notation of h*-closures.
We already have Theorem 1.7 that characterizes the finite-degree Z*-closures. Further, since the finite-degree Z-closure of a symmetric set is symmetric, we have Z-cl n, With an additional observation, we will conclude the following.
Further, using Observation 1.9, Theorem 1.10 and a tight construction of hyperplane cover, we will prove Theorem 1.5, our solution to Problem 1.2 (a) in the Boolean cube setting.
It must be noted that for larger uniform grids, for a weight-determined set, the finite-degree h-closure and Z-closure need not be equal.
We do not yet know how to approach the finite-degree hyperplane closures for larger uniform grids. We, therefore, have the following open questions.
The following fact is folklore and follows easily from the definitions. (See, for instance, Nie and Wang [NW15, Proposition 5.2] for a proof.) It connects affine Hilbert functions with finite-degree Z-closures.
We remark that using Fact 1.13 and the result of Bernasconi and Egidi (Theorem 1.12), we could obtain Theorem 1.10, that is, our combinatorial characterization of finite-degree Z*-closures (as well as finite-degree Z-closures and h-closures) of symmetric sets of the Boolean cube. However, our arguments, in fact, prove Theorem 1.7, that is, our proof works over general uniform grids and could also be regarded as being more combinatorial.

Other applications
We believe that the combinatorial characterization in Theorem 1.7 might also be of independent interest. Indeed, we give a couple of other applications.

An alternate proof of a lemma by Alon et al. [ABCO88]
Alon, Bergmann, Coppersmith and Odlyzko [ABCO88] obtained a tight lower bound for a balancing problem on the Boolean cube {−1, 1} n . Their proof is via the polynomial method, using the following lemma.
Lemma 1.14 ([ABCO88]). Let n ∈ Z + be even and f (X) ∈ R[X] represent a nonzero function on {−1, 1} n such that either of the following conditions is true.

Certifying degrees of weight-determined sets
Let G be a uniform grid. We say a polynomial f (X) ∈ R[X] is a certifying polynomial for a subset The notion of a certifying polynomial (in the Boolean cube setting, for Boolean functions and Boolean circuits) was studied by Kopparty and Srinivasan [KS18] to prove lower bounds for a certain class of Boolean circuits. Variants of this notion have appeared in theoretical computer science, specifically in complexity theory literature, in the works of Aspnes, Beigel, Furst and Rudich [ABFR93], Green [Gre00], and Alekhinovich and Razborov [AR01]. Certifying polynomials have also appeared in the context of cryptography in Carlet, Dalai, Gupta and Maitra [CDGM06].
For a subset S ⊆ G, the certifying degree of S, denoted by cert-deg(S), is defined to be the smallest d ∈ [0, N ] such that S has a certifying polynomial with degree at most d. We will determine the certifying degrees of all weight-determined sets, when G is an SU 2 grid. In fact, we will get the following.
We will conclude our work by considering a third variant of our covering problems -the exact covering problem. As for the other covering problems, we will define (in Section 6) an exact hyperplane (polynomial) cover for a weight-determined set E, E [0, N ] in a uniform grid G, and consider the minimum size (degree) of an exact hyperplane (polynomial) cover for E. Let us denote these by EHC G (E) and EPC G (E) respectively. (In the Boolean cube setting, we will denote these by EHC n (E) and EPC n (E) respectively.) We will see some partial results and conjecture a solution in the Boolean cube setting.
We can quickly indicate the progress made in our work via the following tables.

Cover
Nontrivial Proper Exact  In this work, we have obtained combinatorial characterizations of the quantities in the colored cells in the above tables. Further, the cells with the same color have the same characterization, that is, although the corresponding questions are different, the answers are the same. Characterizations of the quantities in the uncolored (white) cells is open.
Digression 1: Why consider only uniform grids? It is easy to see that all the results in this work that hold for a uniform grid G would also hold for the image of G under any invertible affine transformation of R n . A typical grid that is genuinely nonuniform would be of the form are not equal, for some j ∈ [n]. In this case, an obvious way to define the weight of an element s j for all j ∈ [n]. 7 Unlike in the uniform setting, it is no longer true that every weight-determined set w, w ∈ [0, N ] is contained in the zero set of the linear polynomial j∈[n] X j − w. This has undesirable ripple effects; for instance, the finitedegree Z*-closures of weight-determined sets depend on the weights of the points, as well as on the coordinates of the points in the set.
If the grid is uniform, however, the finite-degree Z*-closures of weight-determined sets depend only on the weights of the points in the set. So the upshot is that the 'uniform' condition on grids ensures the setting is nice enough for our tools, tricks and techniques to work well, and give neat results.
Digression 2: What happens over fields other than R? Consider the Boolean cube {0, 1} n , as well as any uniform grid G ⊆ N n . A crucial observation that enables our algebraic arguments (over {0, 1} n as well as over G) to go through is that for any valid w ∈ N, the zero set of the linear polynomial i∈[n] X i − w is exactly w (in {0, 1} n as well as in G). This is true over any field of characteristic zero. Indeed, all the results in this paper hold true over any field of characteristic zero, without any change in the arguments. Further, it is easy to see that the above mentioned property is also true, and therefore all the results in this paper also hold true, over any field with large positive characteristic p: we require p > n over {0, 1} n , and p > N over G. However, our results do not extend to fields with small positive characteristic.
Organization of the paper. In Section 2, we will look at some preliminaries concerning the different closure operators of interest to us. In Section 3, we will prove our combinatorial characterization of Z*-closures of weight-determined sets of an SU 2 grid, and then obtain Theorem 1.4, our more refined solution to Problem 1.2 (b) for SU 2 grids. In Section 4, we will see that the h-closures coincide with the Z-closures and Z*-closures, for all symmetric sets of the Boolean cube. We also obtain Theorem 1.5, our solution to Problem 1.2 (a) in the Boolean cube setting. In Section 5, we consider further applications of our combinatorial characterization of finite-degree Z*-closures from Section 3: (i) an alternate proof of a lemma by Alon et al. (1988) in the context of balancing problems, and (ii) a characterization of the certifying degrees of weight-determined sets. We conclude in Section 6 by introducing a third variant of our covering problems. We will discuss some partial results and conjecture a solution in the Boolean cube setting.
We adopt the convention that deg (0)   When G = {0, 1} n , it follows that the finite-degree Z-closure of a symmetric set is symmetric. However, for a general uniform grid G, the finite-degree Z-closure of a weight-determined set need not be weight-determined. We recall the one-to-one correspondence between weight-determined sets of G and subsets of [0, N ]. Every subset E ⊆ [0, N ] corresponds to the weight-determined set E = {x ∈ G : wt(x) ∈ E}. For convenience, we will identify E with E. We therefore consider a new notion of closure exclusiely for weight-determined sets. Let G be a uniform grid.
The following properties of finite-degree Z*-closures are similar to that of finite-degree Z-closures, and follow quickly from the definition.
(b) Note that, as a set operator, we have Z*-cl G,d : W(G) → W(G), where W(G) denotes the collection of all weight-determined sets of G. We observe the following. For our results, we are most interested in the setting where the uniform grid G is strictly unimodal (SU 2 ), that is, we have There is a simple characterization of SU 2 grids given by Dhand [Dha14]. (k i − 1) ≤
3 Finite-degree Z-closures and Z*-closures, and our polynomial covering problems In this section, we will obtain our combinatorial characterization of finite-degree Z*-closures of weight-determined sets, and then proceed to solve Problem 1.2. We begin with some simple results. Let G be a uniform grid. The following fact is folklore and follows, for instance, from the Footprint bound (see Cox, Little and O'Shea [CLO15, Chapter 5, Section 3, Proposition 4]). The following result is elementary.

Two main lemmas
We require two main lemmas to obtain our characterization; let us prove them here.
Our first lemma holds over any uniform grid and identifies a collection of 'layers' which are certain to lie in the finite-degree Z*-closures of weight-determined sets. Proof. Let m = min E, M = max E. Consider any j ∈ [0, m − 1]. Suppose j ∈ Z*-cl G,d (E). This means j ⊆ Z-cl G,d (E). So there exists a ∈ j and P (X) ∈ R[X] such that deg P ≤ d, P | E = 0 and P (a) = 1. Define Then we have deg Proof. Note that Note that in this system, the variables are c γ , γ ∈ [0, i], and the constraints are indexed by a ∈ T ′ N,i . So the number of variables is , and therefore this system has a nontrivial solution. Thus, there exists a nonzero polynomial P (X) ∈ span{X γ : γ ∈ [0, i]} such that P | T ′ N,i = 0. By the Combinatorial Nullstellensatz (Theorem 2.1), we then get P = 0 in V (G), that is, P (a) = 0 for some a ∈ G. (Also, obviously by the spanning set description, we get deg

The main theorem
We will now prove Theorem 1.7, our combinatorial characterization of finite-degreee Z*-closures of all weight-determined sets in an SU 2 grid. We restate Theorem 1.7 for convenience. So now assume that |E| ≥ d + 1. By the Closure Builder Lemma 3.3 and Observation 3.4, we already have L N,d (E) ⊆ Z*-cl G,d (E). Now consider any j ∈ L N,d (E). In particular, L N,d (E) = [0, N ]. We will prove that j ∈ Z*-cl G,d (E). Let Note that

Let us now show that
. So in particular, we have i 0 ∈ L N,d (E), which is a contradiction since we assumed Then deg Q ≤ d − i 0 and Q(x 0 ) = 0. This gives deg P Q ≤ d, P Q| E = 0 and P Q(x 0 ) = 0, that is, . This completes the proof.

Computing the finite-degree Z*-closures efficiently
The characterization in Theorem 1.7 gives a simple algorithm to compute Z*-cl G,d (E), for any . We only consider the complexity of computing the finite-degree Z*-closures modulo the bit complexity of representing the weight-determined sets in the uniform grid G; accomodating the bit complexity will only multiply our bound by a poly(log N ) factor. We will now describe the algorithm.
We Immediately, we have the following analogue of Observation 3.4.
The following properties of the above set operators will enable us to compute them in linear time.
Proof. Items (a) and (b) , b}), for all k ∈ N. We will show this by induction on k. The case k = 0 is obvious, since L 0 [a,b],d is the identity operator. Now suppose the claim is true for some k ∈ N.
where the first equality is by the definition of L [a,b],d , and the second equality is by the induction hypothesis. But we already have So from (1) and (2), we get This completes the proof.
We then get a straightforward linear time recursive algorithm to compute . The base case of the recursion appeals to Proposition 3.7 (a) and (b), and the recursive step appeals to Proposition 3.7 (c). The linear run-time is obvious since it is easy to see that there exists a constant C > 0 such that for any a, b ∈ Z, a ≤ b and d ∈ We conclude by stating the pseudocode for this algorithm. Here we assume that the input E ⊆

Solving our polynomial covering problems
Let us gather the work done so far to solve our polynomial covering problems (Problem 1.2 (b)). Let us begin by proving Lemma 1.6 that relates our polynomial covering problems with the finite-degree Z*-closures.
Proof of Lemma 1.6. Let G be a uniform grid and consider E [0, N ].
(a) We have the following equivalences.
Let us now proceed to prove Proposition 1.8. We require the following lemma.
We clearly have the following implications. and We are now ready to prove Proposition 1.8.
We have thus proved Theorem 1.4, which is our solution to Problem 1.2 (b) for SU 2 grids.

Finite-degree h-closures and our hyperplane covering problems
We introduce another new closure operator, defined using polynomials representing hyperplane covers. Let Let G be a uniform grid. For any d ∈ [0, N ] and S ⊆ G, we define the degree-d h-closure of S as h-cl G,d (S) = Z(I(H n , S) d ). We will focus on the case of the Boolean cube. It is immediate that the finite-degree h-closure of a symmetric set is symmetric, and so we will use our indentification of symmetric sets of {0, 1} n with subsets of [0, n].
Our main result in this section is a characterization of finite-degree h-closures of all symmetric sets of the Boolean cube. In fact, we prove that these coincide with the finite-degree Z-closures (and Z*-closures). Let us first show that Lemma 3.3 and Lemma 3.5 have analogues for finite-degree hyperplane closures.
Proof. (a) The proof is similar to that of Lemma 3.3; instead of considering polynomials in R[X], we just need to consider polynomials in H n throughout. (b) We only need to consider i ≤ min{d, ⌊n/2⌋}. The other case can be argued exactly as in the proof of Lemma 3.5. Now consider the polynomial such that x 2t−1 = x 2t = 1; this gives P (x) = 0. So P | T n,i = 0. Now consider any j ∈ [i, n − i].
Hence h-cl n,d (T n,i ) = T n,i .
We can now prove Theorem 1.10.
Proof of Theorem 1.10. The proof is similar to that of Theorem 1.7. Instead of considering polynomials in R[X], we need to consider polynomials in H n throughout. In addition, we need to replace  By Observation 1.9 (b), Theorem 1.10, Proposition 1.8 (b) and Observation 4.2, we have proved Theorem 1.5 (b). This completes our solution to Problem 1.2 (a) in the Boolean cube setting.

Other applications
Our combinatorial characterization of finite-degree Z*-closures of weight-determined sets in SU 2 grids (Theorem 1.7) may also be interesting in its own right. Indeed, we will consider two other applications in this section.

An easy proof of a lemma by Alon et al. [ABCO88]
Consider the following simple fact; the proof is obvious.  Appealing to Fact 5.1, Lemma 1.14 states that Z*-cl n,n/2−1 (E 0 ) = Z*-cl n,n/2−1 (E 1 ) = [0, n]. This is equivalent to Proposition 1.15, by Theorem 1.7. In fact, we can prove the following slightly more general statement; Proposition 1.15 is a special case. For simplicity, let us just prove Proposition 5.2 for the case of N being an even positive integer, and m = 2. The general case can be proven along similar lines.
Our argument is an illustration of Algorithm 1, by using Proposition 3.7. Recall the set operators It is easy to see that for each i ∈ [k], we have We will prove, by induction, that L [k−i,k+i],i−1 (F i ) = [k − i, k + i], for all i ∈ [k]. By Proposition 3.7 (a), we get the base case as . So by Proposition 3.7 (c) and the induction hypothesis, we get This completes the proof.

Certifying degrees of weight-determined sets
Recall that for a uniform grid G and subset S ⊆ G, the certifying degree cert-deg(S) is defined to be the smallest d ∈ [0, N ] such that S has a certifying polynomial with degree at most d. • a polynomial P (X) ∈ R[X] is an exact polynomial cover of E if E = Z G (P ).
Let EHC G (E) and EPC G (E) denote the minimum size of an exact hyperplane cover and the minimum degree of an exact polynomial cover respectively, for a weight-determined set E, E [0, N ]. In the case G = {0, 1} n , we will instead use the notations EHC n (E) and EPC n (E).
We first note that EPC G can be characterized in terms of the finite-degree Z-closures. Contrast this with Lemma 1.6 which characterizes PPC G in terms of the finite-degree Z*-closures. Proof of Proposition 6.1. Let d ′ = EPC G (E) and d ′′ = min{d ∈ [0, N ] : Z-cl G,d (E) = E}. There exists a polynomial P (X) ∈ R[X] such that deg P = d ′ , P | E = 0 and P (a) = 0, for all a ∈ G \ E. This implies Z-cl G,d ′ (E) = E, and so d ′′ ≤ d ′ .
Further, for every a ∈ G \ E, there exists Q a (X) ∈ R[X] such that deg Q a ≤ d ′′ , Q a | E = 0 and Q a (a) = 1. We can then choose scalars β a ∈ R, a ∈ G \ E such that the polynomial Q(X) := a∈G\E β a P a (X) satisfies deg P ≤ d ′′ , P | E = 0 and P (a) = 0, for all a ∈ G \ E. So d ′ ≤ d ′′ , and this completes the proof. However, we do not have a further characterization of the finite-degree Z-closures of weightdetermined sets. In the Boolean cube setting, however, since the finite-degree Z-closures and Z*closures coincide for all symmetric sets, we immediately get the following by appealing to Theorem 1.4, Theorem 1.5, Theorem 1.7 and Theorem 1.10. Further, characterizing EHC n (E) for E [0, n] seems to be even more difficult. We have the following partial results.

Further remarks: Some updates post publication
This section was added post the publication of this work. Here, we mention a few remarks that complete this work better. 7.1 The exact covering problem: Conjecture 6.4 is false, and EHC n = EPC n Consider the Boolean cube {0, 1} n . We will characterize EHC n (E), for all E [0, n]. By the definitions and Corollary 6.2, we get EHC n (E) ≥ EPC n (E) = PPC n (E) = PHC n (E) = |E| − max{i ∈ [0, n] : T n,i ⊆ E}.
In Proposition 6.3 (a) and (b), we see that (for all valid n) the first inequality above is tight in the cases (i) T n,1 ⊆ E, and (ii) T n,1 ⊆ E, T n,2 ⊆ E. The only remaining case is T n,2 ⊆ E. In this case, Proposition 6.3 (c) and Conjecture 6.4 respectively state that the first inequality is tight if E = T n,2 , and not tight otherwise. Now, we show that the first inequality is tight if T n,2 E. In particular, Conjecture 6.4 is false.
In a recent work, Ghosh, Kayal, and Nandi [GKN22] gave the following construction of hyperplanes, while solving a different hyperplane covering problem. This construction, in fact, extends our construction given in the proof of Proposition 6.3 (c). We state this in our notation, which agrees with [GKN22] up to a change of coordinates. X t − j.
Then {H j : j ∈ [0, i − 1]} is an exact hyperplane cover of T n,i , having size i.
We then immediately get the following. So the progress made in our work, as indicated in Table 1, can be updated to the following.

Cover
Nontrivial Proper Exact Hyperplane HC n PHC n EHC n Polynomial PC n PPC n EPC n

Tight examples
Here, for convenience, we quickly collect all the tight examples to the problems that we have solved.
(a) HC n (E) = PC n (E). Let d 0 = min{d ∈ [0, n] : E is d-admitting}, and i 0 = max{i ∈ [0, d 0 ] : E is (d 0 , i 0 )-admitting}. Then we have E ∪ T n,i 0 = [0, n], and necessarily, |E \ T n,i 0 | = d 0 − i 0 . A tight example is (2) Let G be an SU 2 grid, and consider any E Acknowledgements. The author thanks • his graduate advisor Srikanth Srinivasan for valuable discussions throughout the gestation of this work, and unending support and encouragement.
• Niranjan Balachandran for helpful comments on a preliminary version of this work.
• Murali K. Srinivasan for a very enlightening discussion, as well as useful suggestions on the presentation of this work.
• Lajos Rónyai for pointers to some relevant literature.
• Anurag Bishnoi for narrating the history of the hyperplane covering problems and the polynomial method, as well as for pointing out the recent work [GKN22].
• the anonymous referee for critical comments in an eagle-eyed review.